Teaching the Concept of Limit by Using Conceptual Conflict ...

Teaching the Concept of Limit by Using Conceptual Conflict Strategy and Desmos Graphing Calculator

Senfeng Liang University of Wisconsin-Stevens Point, U.S.A., liangsenfeng@



To cite this article: Liang, S. (2016). Teaching the concept of limit by using conceptual conflict strategy and Desmos graphing calculator. International Journal of Research in Education and Science (IJRES), 2(1), 35-48.

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International Journal of Research in Education and Science

Volume 2, Issue 1, Winter 2016

ISSN: 2148-9955

Teaching the Concept of Limit by Using Conceptual Conflict Strategy and Desmos Graphing Calculator

Senfeng Liang* University of Wisconsin-Stevens Point, U.S.A.

Abstract

Although the mathematics community has long accepted the concept of limit as the foundation of modern Calculus, the concept of limit itself has been marginalized in undergraduate Calculus education. In this paper, I analyze the strategy of conceptual conflict to teach the concept of limit with the aid of an online tool ? Desmos graphing calculator. I also provide examples of how to use the strategy of conceptual conflict. This graphing calculator provides an interactive, dynamic, and persuasive approach of teaching limit. I focus on applying the

conceptual conflict idea to the concept of limit in the situation where x approaches infinity. This strategy can be applied to the limit of a function when x approaches a fixed number.

Key words: Calculus; Conceptual change; Conceptual conflict; Limit; Technology

Introduction

Although the mathematics community has long accepted the concept of limit as the foundation of modern Calculus, the concept of limit itself has been marginalized in undergraduate Calculus education. Textbooks have

paid little attention to its fundamental aspects ? the relationship and teachers treat this topic hurriedly

and cannot wait to move to next topics (Bokhari & Yushau, 2006). For non-mathematics majors, lack of understanding of the concept of limit may not be a serious problem for their study of further mathematics courses; for mathematics majors, however, it is an important issue. Students' understandings of Calculus will greatly influence their ability to study more advanced Analysis courses (such as Real Variable Function, Real Analysis, Functional Analysis, and Measure Theory) because these courses all require Calculus as a prerequisite. There are many sources that contribute to the difficulties of teaching and learning this concept (Cornu, 1992; Davis & Vinner, 1986). Nevertheless, teaching the concept of limit successfully is not an unattainable task if we use proper strategies and tools. In this paper, I analyze the strategy of conceptual conflict for teaching the concept of limit with the aid of an online Desmos graphing calculator. I focus on applying the

conceptual conflict idea to the concept of limit in the situation where x approaches infinity. This strategy can be easily applied to the limit of sequences and limit of a function when x approaches a fixed number. The idea

of conceptual conflict can be generalized to other situations in Calculus, such as teaching the concepts of the continuity and derivative.

Theoretical Framework

A principal tenet of constructivist learning theory is that new knowledge (or concept) builds on prior knowledge (or concept) (Davis & Vinner, 1986; Hewson & Hewson, 1984; Meyer, 1993). Conceptual conflict is the conflict between the new concept with the learner's prior concept. The process of learning through connecting previous knowledge to new knowledge is often related to conceptual change. Vosniadou (2007) defines conceptual change:

In order to understand the advanced scientific concepts of the various disciplines, students cannot rely on the simple memorization of facts or the enrichment of their naive, intuitive theories. They need to be able to restructure their prior knowledge which is based on everyday experience and lay culture, a restructuring that is known as conceptual change. (p. 47-48)

* Corresponding Author: Senfeng Liang, liangsenfeng@

36 Liang

When the connection between previous knowledge and new knowledge is unsuccessful, errors in understanding (or misconceptions) may appear (Meyer, 1993). For example, students may apply the experience of multiplication of positive integers to the multiplication of fractions so they may believe that multiplications always make numbers bigger. However, when students learn new knowledge they may not even be aware of the need to change the underlying logic of thinking and their own misconceptions (Merenluoto & Lehtinen, 2000). Although research on conceptual change is mainly conducted in the field of science education, there is increasing attention to this idea in the field of mathematics education (Vamvakoussi & Vosniadou, 2004).

Conceptual change requires students to see the necessities for a change in their concepts (Merenluoto & Lehtinen, 2000). If they do not recognize a need for conceptual change, they may develop misconceptions. For example, when students learn that the complex number system is an extension from real number systems, they may think that it is reasonable to compare any complex numbers because they can do that in the real number system. However, that is not the case for complex system. One cannot tell which one is bigger for any given two given imaginary numbers. In order to prevent and correct students' misconceptions, teachers should apply approaches to achieve students' conceptual change. In the fields of science and mathematics education, researchers have identified several useful strategies for supporting conceptual change: analogies, multirepresentation, discussion, historical perspectives, and conceptual conflicts (Calik, Ayas, & Coll, 2008; Meyer, 1993; Nussbaum & Novick, 1982; Singer, 2007). It is reasonable to use different strategies in different situations. This paper will focus on applying the conceptual conflict approach to teaching the concept of limit.

According to Trumper (1997), if a conceptual change is needed for students, the first step is to let students be aware of the necessity of such a change and feel dissatisfaction with the explanation that is based on previous knowledge. Conceptual conflict is one of the effective ideas to help students to recognize such dissatisfaction (Meyer, 1993). If students fail to feel dissatisfaction, they are not likely to develop a conceptual change spontaneously. However, too often, students are not given the change to feel such dissatisfaction and are merely provided new knowledge by the teacher, new definitions and the classical experiments. According to Nussbaum and Novick (1982), "...students read about, observe or even perform those experiments before they are given the chance to recognize the existence and nature of the problem and they, therefore, do not get to experience any cognitive conflict at all" (p. 186). Although Nussbaum and Novick's (1982) article is mainly about science education, the situation is similar in mathematics education. In mathematics, students are taught by presentations of definitions and concepts rather than developing an understanding of why they need these definitions and concepts. This phenomenon is common in college level mathematics.

The strategy of conceptual conflict provides an approach for students to be aware of the need to make conceptual change (Nussbaum & Novick, 1982). The first step is to let students recognize explicitly their misconceptions--that is, the step of diagnosis. Instead of hastening to introduce new concepts to students, teachers can spend some time to identify students' previous understandings of the intended concepts because students often bring lots of informal mathematics knowledge to classrooms. Then teachers can provide some problems and ask students to solve and explain these problems in the class. Different students may have different solutions and explanations. Based on students' statements of their understanding, teachers will be able to categorize some features of their misconceptions.

At the second step, teachers can lead students to debate the pros and cons of these statements ? that is, the step of clarification. After the debate and arguments of the students, some statements will survive, some will be eliminated, and some will be undecided. At this step, teachers will be able to summarize students' misconceptions: some of them may be particular to a few students and others may be common to many students.

At the third step teachers need to confront students' misconceptions. Teachers can try to find the logical problems within students' replies or present students with counterexamples that violate their misunderstandings. The counterexamples are expected to confront students' misconceptions explicitly. Teachers need to defend why a scientifically agreed-upon conception or a formal definition has more empirical or theoretical validity than students' conceptions. The basic tool is the logic: the formal concept should make sense. Some constructivists may argue it will be better to induce students to develop the new concepts by themselves (Bishop, 1967). But this is not always that case especially for the subjects like mathematics which has subsided for thousands of years with the endeavors of mathematicians. Students are necessarily able to develop the desirable mathematical understanding even if they feel dissatisfaction of their original explanation.

At the last step, it is hopefully that by now students are willing to accept the new knowledge and change their misconceptions ? that is, the step of accommodation. At this step, teachers can help students to incorporate the

International Journal of Research in Education and Science (IJRES) 37

new knowledge to students' previous knowledge because teachers have created the necessary conditions to learn the new knowledge for students (Meyer, 1993).

Application of Conceptual Conflict to Teach the Concept of Limit

Why Limit Matters?

The concept of limit is situated in an ironic place in the current Calculus education. On one side, limit is the fundamental concept for modern Calculus and related subjects such as Measure Theory, Real Analysis, and Functional Analysis. Rarely do mathematicians refuse its fundamental role in Calculus and other Analysis. Students often begin to learn Calculus with the concepts of function and limit. If mathematics majors do not understand the concept of limit, they are not likely to understand the concepts of continuity, uniform continuity, convergence, derivative, and they are not likely be ready to take other Analysis courses. On the other side, the concept of limit has been marginalized in textbooks, teaching, and research. (Bokhari & Yushau, 2006). For example, textbooks usually pay little attention to the theoretical aspects of the concept of limit (Ostebee & Zorn, 2001; Stewart, 2012). Calculus teachers usually focus on the calculation of limit, sometimes on graphical illustration of limit, rarely on theoretical aspect (or definition) of limit.

One of the most important features that differentiate mathematics from other subjects is the upgrade from intuitive concrete understanding to abstract recognition. The concept of limit is such an example. Before students first learn the concept of limit, they already have some experiences of what a limit is (Merenluoto & Lehtinen, 2000). Their understanding is mainly based on everyday experiences rather than mathematical understandings. If students do not change their everyday understanding of limit to mathematical understanding of limit, they will not be able to upgrade from intuitive concrete understanding to abstract recognition. In order to reach the deep understanding of Calculus, curriculum and instrumental design need to be theoretically grounded in the actual mathematics and find an effective approach within the mathematics community. Despite of its importance, students find it difficult to understand the concept of limit. The next highlights the common sources of difficulties in teaching the concept of limit informed through the history of limit in Calculus.

Sources of Difficulties in the Teaching the Concept of Limit

There are many sources that may contribute to the difficulties of teaching the concept of limit. First, the difficulties may come from students' misconceptions of limit. Some researchers have identified some common misconceptions of limit: A sequence "must not reach its limit"; a limit is a boundary beyond which the sequence cannot go; a limit is a stop-sign, etc. (Davis & Vinner, 1986; Merenluoto & Lehtinen, 2000; Tall & Vinner, 1981). For example, the differences between the everyday language and the mathematics language contribute students' misconceptions (Cornu, 1992; Kim, Sfard, & Ferrini-Mundy, 2005). When students enter the Calculus classrooms they bring their everyday experiences and understanding of "limit" with them which though necessary can lead to misconceptions and hence also bring learning obstacles. For example, one may say that my limit of running is two miles, and three hours is my limit to keep continuous working. These everyday understandings of limit suggest that limit is some value one cannot go across.

Sometimes even Calculus book writers may be unable to notice these differences between everyday language and the mathematics language and thus add fuel to fire of students' misconception. For example, in order to explain the concept of limit to his or her student, one book use an example like this: in order to explain the

meaning of x 3 , the authors used an analogy: you can get access infinitely close to a running fan but

obviously you will never reach it because you know what will happen if you reach the running fan (Adams, Thompson, & Hass, 2001). In this case, the book strengthened the misconception that limit is something that you can infinitely approach but never actually reach. The risk of this strategy is that when one person approach a

fan, he usually only approach it from one side but x 3 mean x 3 and x 3 . The lack of awareness

of misconceptions of the concept of limit is one of the main sources of the current situation in both teaching and learning the concept of limit.

Second, students' perspectives on the learning of the concept of limit are another source of difficulty (Cornu, 1992; Szydlik, 2000; Tall & Vinner, 1981). Besides lack of recognition of misconception, most students also do not see the value of understanding the concept of limit: their teachers say this is optional; their textbooks place the definitions in the end of chapters; their experience tells them it is often useless in exams. Students and

38 Liang

teachers often use achievement in exams to judge whether or not students have mastered the intended knowledge. This perspective promotes students' belief that the most important thing about limit is to know how to calculate it instead of understanding it and how to prove the existence of limit. Meyer (1993) argued that students who rely exclusively on terminology, memory of text, or common sense may answer exam questions correctly, but they may not understand their answers fully or may still believe falsely (p. 106). Cornu (1992) confirmed that it looked like that lack of understanding the concept of limit is not a problem for students' exams.

Different in investigations which have been carried out show only too clearly that the majority of students do not master the idea of limit, even at a more advanced stage of their studies. This does not prevent them from working out exercises, solving problems and succeeding in the examinations (Cornu, 1992, p. 154).

However, mathematic major Johnson's (2007) experience as indicates how lack of fundamental understanding of the concept of limit and convergence in Calculus hindered her study of Analysis.

Third, the difficulty in teaching the limit concept has a close relationship with purpose of the Calculus course, teachers' attitude, content knowledge and everyday instruction. Because typically both mathematics majors and non-mathematics majors take the same Calculus courses together, it is unlikely that the instructor would apply a rigorous treatment for this course. However, this may cause serious problem for mathematics majors thereafter.

On one hand, it is possible that some Calculus teachers themselves do not properly understand the concept of limit or have misconceptions of this concept. This is evident through the study by Mastorides and Zachariades' (2004) in which they indicated teachers' content knowledge of limit was incomplete and that it affected the pedagogical content knowledge. For example, "most of them have difficulties in understanding multi-quantified statements or fail to comprehend the modification of such statements brought about by changes in the order of the quantifiers (pp. 481-488)." Whether or not this happen to the college level Calculus teaching needs further investigation. On the other hand, some teachers may understand the concept of limit very well, but they are likely to think it is too challenging for students to learn so they do not emphasize this topic and just skip it.

Teachers take the decision to present or exclude the definition in their classes and some tell their

students to memorize this definition; it will take them some time to absorb it. Many instructors put little emphasis on its explanation and look for an appropriate instance for its swift presentation before moving to the next topic. This may be one of the reasons that the definition does not hold a unified position in basic Calculus book (Bokhari and Yushau, 2006, p. 156).

Calculation and conceptual understanding are both important in the Calculus teaching. However, most emphasis

has been placed on how to calculate the limit instead of on understanding its definition. Sometimes, proving the

existence of limit is as important as finding the limit. For example, if students do not see the necessity to prove

the existence of lim(1 1 )x they are not likely to find its limit because e is actually defined by this limit!

x

x

When students take advanced Analysis classes they will feel a real need to understand the definitions of limit

and the proofs of theorems related to the way of thinking. This has been confirmed by Johnson (2007).

She argued that when she took Calculus, she did not understand why it was necessary to prove that a sequence for which one has calculated a limit indeed has a limit. After she took analysis, she noticed that

...the nature of this course (analysis) was different from ones I had taken in the past, but none of these differences were stated or clarified. The fact that the professor paid no attention to the differences in the nature of the mathematics courses made learning the material a difficult chore (Johnson, 2007, p. 286).

Fourth, textbooks' treatment of the concept of limit has a deep influence on teachers' instruction (Bokhari and Yushau, 2006; Cornu, 1992). Textbooks usually use "nice" and "good" examples when introducing new concepts (Gruenwald & Klymchuk, 2003). This holds for the concept of limit in particular (Gruenwald &

Klymchuk, 2003). For example, at the sections about the limit of a function when x approaches infinity, two

textbooks use the example y 1 for introduction (Ellis & Gulick, 2002; Ostebee & Zorn, 2001). The using of x

"nice" examples are pedagogically helpful but also risks of promoting students' misconceptions because it gives

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